Problem: $\dfrac{d}{dx}(5x^4-2x^3-x^2)=$
Answer: According to the sum rule, the derivative of $5x^4-2x^3-x^2$ is the sum of the derivatives of $5x^4$, $-2x^3$, and $-x^2$. The derivatives of these terms can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ For example, this is the derivative of the first term: $\begin{aligned}\dfrac{d}{dx}(5x^4)&=5\dfrac{d}{dx}(x^4)&&\gray{\text{Constant multiple rule}}\\\\ &=5\cdot (4x^3)&&\gray{\text{Power rule}}\\ \\ &=20x^3\end{aligned}$ Here is the complete differentiation process: $\begin{aligned} &\phantom{=}\dfrac{d}{dx}(5x^4-2x^3-x^2) \\\\ &=5\dfrac{d}{dx}(x^4)-2\dfrac{d}{dx}(x^3)-\dfrac{d}{dx}(x^2)&&\gray{\text{Basic differentiation rules}} \\\\ &=5\cdot4x^3-2\cdot3x^2-2x&&\gray{\text{The power rule}} \\\\ &=20x^3-6x^2-2x \end{aligned}$ In conclusion, $\dfrac{d}{dx}(5x^4-2x^3-x^2)=20x^3-6x^2-2x$.